Question

Find the value of k for which the system of equations kx + 3y = 4 and 3x − 9y = −12 has a unique solution.

Answer 1

Note:

If we consider two linear equations in two variables say:

a1x + b1y + c1 = 0 and

a2x + b2y + c2 = 0.

Then, the condition for the unique solution is: a1/a2 ≠ b1/b2

Here, the given equations are;

kx + 3y = 4 and 3x - 9y = - 12

ie,

kx + 3y - 4 = 0 and

3x - 9y + 12 = 0

Clearly,

Here we have;

a1 = k

a2 = 3

b1 = 3

b2 = - 9

c1 = -4

c2 = 12

Thus, the given system of linear equations would have unique solution if and only if;

=> a1/a2 ≠ b1/b2

=> k/3 ≠ 3/-9

=> k ≠ -1

Thus, k can have any real value except

(-1).